Eigenvalues of Inhomogenous Structures

Instead of the cover......Page 1
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Other Books from Professor Isaac Elishakoff......Page 6
Contents......Page 7
Foreword......Page 15
Prologue......Page 17
1.1 Introductory Remarks......Page 23
1.2 Vibration of Uniform Homogeneous Beams......Page 24
1.3 Buckling of Uniform Homogeneous Columns......Page 26
1.4 Some Exact Solutions for the Vibration of Non-uniform Beams......Page 35
1.4.1 The Governing Differential Equation......Page 37
1.5 Exact Solution for Buckling of Non-uniform Columns......Page 40
1.6 Other Direct Methods (FDM, FEM, DQM)......Page 44
1.7 Eisenberger's Exact Finite Element Method......Page 46
1.8 Semi-inverse or Semi-direct Methods......Page 51
1.9 Inverse Eigenvalue Problems......Page 59
1.10 Connection to the Work by Zvczkowski and Gajewski......Page 66
1.11 Connection to Functionally Graded Materials......Page 68
1.12 Scope of the Present Monograph......Page 69
2.1.1 Introductory Remarks......Page 71
2.1.2 Formulation of the Problem......Page 72
2.1.3 Uncovered Closed-Form Solutions......Page 73
2.2.2 Formulation of the Problem......Page 81
2.2.3 Column Pinned at Both Ends......Page 82
2.2.4 Column Clamped at Both Ends......Page 84
2.2.5 Column Clamped at One End and Pinned at the Other......Page 85
2.2.6 Concluding Remarks......Page 86
2.3.1 Introductory Remarks......Page 90
2.3.2 Formulation of the Problem......Page 92
2.3.4 Column Pinned at One End and Clamped at the Other......Page 95
2.3.5 Column Clamped at One End and Free at the. Other......Page 97
2.3.6 Concluding Remarks......Page 99
2.4.1 Introductory Remarks......Page 100
2.4.2 Basic Equations......Page 103
2.4.3 Column Pinned at Both Ends......Page 108
2.4.4 Column Clamped at Both Ends......Page 113
2.4.5 Column that is Pinned at One End and Clamped at the Other......Page 116
2.4.6 Concluding Remarks......Page 121
3.1.1 Introductory Remarks......Page 123
3.1.2 Formulation of the Problem......Page 124
3.1.3 Inhomogeneous Rods with Uniform Density......Page 125
3.1.4 Inhomogeneous Rods with Linearly Varying Density......Page 128
3.1.5 Inhomogeneous Rods with Parabolically Varying Inertial Coefficient......Page 130
3.1.6 Rod with General Variation of Inertial Coefficient ($m > 2$)......Page 131
3.1.7 Concluding Remarks......Page 134
3.2.1 Introductory Remarks......Page 136
3.2.2 The Nodal Parameters......Page 137
3.2.3 Mode with One Node: Constant Inertial Coefficient......Page 140
3.2.4 Mode with Two Nodes: Constant Density......Page 143
3.2.5 Mode with One Node: Linearly Varying Material Coefficient......Page 145
3.3 Concluding Remarks......Page 147
4.1.1 Introductory Remarks......Page 151
4.1.2 Formulation of the Problem......Page 152
4.1.3 Boundary Conditions......Page 153
4.1.4 Expansion of the Differential Equation......Page 154
4.1.5 Compatibility Conditions......Page 155
4.1.6 Specified Inertial Coefficient Function......Page 156
4.1.7 Specified Flexural Rigidity Function......Page 157
4.1.8 Stochastic Analysis......Page 160
4.1.10 Concluding Remarks......Page 167
4.2.1 Introductory Remarks......Page 168
4.2.2 Formulation of the Problem......Page 169
4.2.3 Cantilever Beam......Page 170
4.2.4 Beam that is Clamped at Both Ends......Page 179
4.2.5 Beam Clamped at One End and Pinned at the Other......Page 181
4.2.6 Random Beams with Deterministic Frequencies......Page 184
4.3.1 Introductory Remarks......Page 191
4.3.2 Basic Equation......Page 196
4.3.3 A Beam with Constant Mass Density......Page 198
4.3.4 A Beam with Linearly Varying Mass Density......Page 201
4.3.5 A Beam with Parabolically Varying Mass Density......Page 206
4.4 Concluding Remarks......Page 215
5.1.1 Introductory Remarks......Page 219
5.1.2 Choosing a Pre-selected Mode Shape......Page 220
5.1.3 Buckling of the Inhomogeneous Column under an Axial Load......Page 221
5.1.4 Buckling of Columns under an Axially Distributed Load......Page 225
5.1.5 Concluding Remarks......Page 240
5.2.1 Introductory Comments......Page 241
5.2.2 Formulation of the Problem......Page 242
5.2.3 Basic Equations......Page 243
5.2.4 Constant Inertial Coefficient ($m = 0$)......Page 244
5.2.5 Linearly Varying Inertial Coefficient ($m = 1$)......Page 246
5.2.6 Parabolically Varying Inertial Coefficient ($m = 2$)......Page 247
5.2.7 Cubic Inertial Coefficient ($m = 3$)......Page 252
5.2.8 Particular Case $m = 4$......Page 255
5.2.9 Concluding Remarks......Page 258
6.1.1 Introductory Remarks......Page 265
6.1.2 Construction of Postulated Mode Shapes......Page 266
6.1.3 Formulation of the Problem......Page 267
6.1.4 Closed-Form Solutions for the Clamped-Free Beam......Page 268
6.1.5 Closed-Form Solutions for the Pinned-Clamped Beam......Page 287
6.1.6 Closed-Form Solutions for the Clamped-Clamped Beam......Page 305
6.1.7 Concluding Remarks......Page 324
7.1.1 Introductory Remarks......Page 325
7.1.3 Boundary Conditions......Page 326
7.1.4 Solution of the Differential Equation......Page 327
7.1.5 The Degree of the Material Density is Less than Five......Page 328
7.1.6 General Case: Compatibility Conditions......Page 334
7.2.1 Introductory Remarks......Page 338
7.2.3 Boundary Conditions......Page 339
7.2.4 Solution of the Differential Equation......Page 340
7.2.5 Cases of Uniform and Linear Densities......Page 341
7.2.6 General Case: Compatibility Condition......Page 343
7.2.7 Concluding Remarks......Page 345
7.3.2 Formulation of the Problem......Page 346
7.3.3 Constant Inertial Coefficient ($m = 0$)......Page 348
7.3.4 Linearly Varying Inertial Coefficient ($m = 1$)......Page 349
7.3.5 Parabolically Varying Inertial Coefficient ($m = 2$)......Page 351
7.3.6 Cubically Varying Inertial Coefficient ($m = 3$)......Page 353
7.3.7 Coefficient Represented by a Quartic Polynomial ($m = 4$)......Page 354
7.3.8 General Case......Page 356
7.3.9 Particular Cases Characterized by the Inequality $n \geq m + 2$......Page 365
7.4.2 Formulation of the Problem......Page 380
7.4.3 General Case......Page 382
7.4.4 Constant Inertial Coefficient ($m = 0$)......Page 392
7.4.5 Linearly Varying Inertial Coefficient ($m = 1$)......Page 393
7.4.6 Parabolically Varying Inertial Coefficient ($m = 2$)......Page 394
7.4.7 Cubically Varying Inertial Coefficient ($m = 3$)......Page 396
7.4.8 [nertial Coefficient Represented as a Quadratic ($m = 4$)......Page 401
7.4.9 Concluding Remarks......Page 408
8.1.1 Introductory Comments......Page 411
8.1.2 Basic Equations......Page 413
8.1.3 Column that is Clamped at One End and Free at the Other......Page 414
8.1.4 Column that is Pinned at its Ends......Page 418
8.1.5 Column that is clamped at its ends......Page 423
8.1.6 Column that is Pinned at One End and Clamped at the Other......Page 427
8.1.7 Concluding Remarks......Page 432
8.2.1 Introductory Comments......Page 433
8.2.2 Formulation of the Problem......Page 435
8.2.3 Basic Equations......Page 437
8.2.4 Closed-Form Solution for the Pinned Beam......Page 438
8.2.5 Closed-Form Solution for the Clamped-Free Beam......Page 447
8.2.6 Closed-Form Solution for the Clamped-Clamped Beam......Page 458
8.2.7 Closed-Form Solution for the Beam that is Pinned at One End and Clamped at the Other......Page 468
8.2.8 Concluding Remarks......Page 476
9.1.1 Introductory Remarks......Page 477
9.1.2 Formulation of the Problem......Page 478
9.1.3 Beam with Uniform Inertial Coefficient, Inhomogeneous Elastic Modulus and Elastic Foundation......Page 479
9.1.4 Beams with Linearly Varying Density, Inhomogeneous Modulus and Elastic Foundations......Page 484
9.1.5 Beams with Varying Inertial Coefficient Represented as an iiith Order Polynomial......Page 491
9.1.6 Case of a Beam Pinned at its Ends......Page 496
9.1.7 Beam Clamped at the Left End and Free at the Right End......Page 502
9.1.8 Case of a Clamped-Pinned Beam......Page 507
9.1.9 Case of a Clamped-Clamped Beam......Page 512
9.1.10 Case of a Guided-Pinned Beam......Page 517
9.1.11 Case of a Guided-Clamped Beam......Page 526
9.1.12 Cases Violated in Eq. (9.99)......Page 531
9.1.13 Does the Boobnov-Galerkin Method Corroborate the Unexpected Exact Results?......Page 533
9.1.14 Concluding Remarks......Page 537
9.2.2 Basic Equations......Page 538
9.2.3 Uniform Inertial Coefficient......Page 539
9.2.4 Linear Inertial Coefficient......Page 542
9.3.1 Introductory Remarks......Page 544
9.3.2 Basic Equations......Page 545
9.3.3 Constant Inertial Coefficient......Page 547
9.3.4 Linear Inertial Coefficient......Page 549
10.1.1 Introductory Remarks......Page 553
10.1.2 Basic Equations......Page 554
10.1.3 Buckling of Non-uniform Pinned Columns......Page 555
10.1.4 Buckling of a Column under its Own Weight......Page 558
10.1.5 Vibration Mode of a Uniform Beam as a Buckling Mode of a Non-uniform Column......Page 560
10.1.6 Non-uniform Axially Distributed Load......Page 561
10.1.7 Concluding Remarks......Page 563
10.2.1 Introductory Comments......Page 564
10.2.2 Evaluation of the Example by Birger and Mavliutov......Page 567
10.2.3 Reinterpretation of the Integral Method for Inhomogeneous Beams......Page 569
10.2.4 Uniform Material Density......Page 571
10.2.5 Linearly Varying Density......Page 573
10.2.6 Parabolically Varying Density......Page 575
10.2.8 Concluding Remarks......Page 579
10.3.1 Introductory Remarks......Page 582
10.3.2 Pinned-Pinned Beam......Page 583
10.3.3 Guided-Pinned Beam......Page 591
10.3.4 Free-Free Beam......Page 598
10.3.5 Concluding Remarks......Page 606
11.1.1 Introductory Remarks......Page 607
11.1.2 Basic Equations......Page 609
11.1.4 Constant Inertial Term ($m = 0$)......Page 610
11.1.5 Linearly Varying Inertial Term ($m = 1$)......Page 611
11.1.6 Parabolically Varying Inertial Term ($m = 2$)......Page 612
11.1.7 Cubic Inertia] Term ($m = 3$)......Page 614
11.1.8 General Inertial Term ($m \geq 4$)......Page 616
11.1.9 Alternative Mode Shapes......Page 617
11.2.1 Introductory Remarks......Page 620
11.2.3 Basic Equations......Page 621
11.3.1 Basic Equations......Page 623
11.3.2 Constant Inertial Term ($m = 0$)......Page 624
11.3.3 Linearly Varying Inertial Term ($m = 1$)......Page 625
11.3.4 Parabolically Varying Inertial Term ($m = 2$)......Page 626
11.3.5 Cubic Inertial Term ($m = 3$)......Page 628
11.3.6 General Inertial Term ($m \geq 4$)......Page 630
11.3.7 Concluding Remarks......Page 632
Epilogue......Page 633
Appendices......Page 643
References......Page 669